Statistics in the Twilight Zone: Component-Specific Inference in Finite Mixture Models

• Activity Number: 67
• Type: Topic Contributed
• Date/Time: Sunday, August 9, 2015 : 4:00 PM to 5:50 PM
• Sponsor: Section on Statistics in Epidemiology
• Author(s): Avi Feller* and Luke W. Miratrix and Natesh Pillai and Evan Greif
• Companies: Harvard University and Harvard University and Harvard University and Harvard University
• Keywords: Mixture models ; Causal inference

Abstract

Component-specific inference in finite mixture models can often yield pathological results, a fact frequently over-looked by applied researchers. For example, many estimators of the component means can yield results that are either degenerate or have the wrong sign. In this paper, we examine from first principles one deceptively simple example: inference for the component means in a two-component Gaussian mixture model with known mixing proportions. We first explore the finite sample properties of various Method of Moments estimators and find the probabilities that these estimators do not exist or yield the wrong sign. We then show that such cases correspond to highly biased maximum likelihood estimates in finite samples. We formalize this idea for one case, finding poor convergence for the MLE when the mixture components are not sufficiently well separated asymptotically. Finally, we demonstrate that these results are relevant for casual inference, especially for model-based principal stratification, which is often implicitly based on component-specific inference in finite mixture models.